Control Limits for an Adaptive Self-Starting Distribution-Free CUSUM Based on Sequential Ranks
Abstract
:1. Introduction
2. Parametric and Nonparametric Univariate CUSUM Control Charts
2.1. Conventional Parametric CUSUM
2.2. Sequential Ranks CUSUM (SRC)
3. Adaptive Control Limit SRC (AC-SRC)
Algorithm 1: Adaptive Control Limit SRC (AC-SRC) |
3.1. Remarks on and Suggestions for the Selection of AC-SRC Parameters
4. Results and Discussion
4.1. Control Limits and Reference Values for the AC-SRC
4.2. Performance Evaluation of the Proposed AC-SRC
4.2.1. Performance under Normality
- Detection delay (DD)
- -
- One of if not the major objective in practical applications is to detect a change as quickly as possible; hence, DD should be small (see also Section 3.1).
- Average run length (ARL)
- -
- Recall that the ARL describes the average time or run length until the control chart signals under in-control conditions, i.e., without a change having occurred. The ARL is, loosely speaking, akin to the type-I error level in hypothesis testing. Rather than setting a false alarm rate control charts are typically designed by choosing a desired . The actual in-control ARL determined in our simulation experiment should be reasonably close to the nominal and we interpret this closeness as indicating the control chart’s robustness.
- False alarm rate (FAR)
- -
- Moreover, recall that even if the monitored process is in-control any CUSUM chart will eventually signal. Clearly there is a relation between FAR and ARL; however, since said relation and false alarm properties of CUSUMs in general are neither well explored nor straightforward, especially for rather small ARLs, a discussion is deemed beyond the scope of this work. The interested reader is referred to, e.g., [28]. Coming back to the issue at hand, as fas as our performance assessment is concerned FAR values should be as small as possible (ideally zero).
4.2.2. Performance under Impulsive Noise Contamination
5. Conclusions
Acknowledgments
Conflicts of Interest
Abbreviations
AC-SRC | Adaptive Control Limit SRC |
ARL | Average Run Length |
CUSUM | Cumulative Sum Control Chart |
DD | Detection Delay |
EWMA | Exponentially Weighted Moving Average Control Chart |
FAR | False Alarm Rate |
SPC | Statistical Process Control |
SRC | Sequential Ranks CUSUM |
Appendix A. Tabularized AC-SRC Control Limits and Reference Values
100 | 100 | 100 | 100 | 100 | 100 | 100 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5486 | 0.5318 | 0.5267 | 0.5209 | 0.5180 | 0.5142 | 0.5131 | |
0.4168 | 0.4274 | 0.4250 | 0.4247 | 0.4221 | 0.4199 | 0.4165 | |
0.8487 | 0.8410 | 0.8331 | 0.8308 | 0.8251 | 0.8209 | 0.8144 | |
1.2013 | 1.2080 | 1.2012 | 1.1910 | 1.1775 | 1.1697 | 1.1598 | |
1.4961 | 1.5056 | 1.4885 | 1.4765 | 1.4627 | 1.4510 | 1.4378 | |
1.7470 | 1.7605 | 1.7395 | 1.7283 | 1.7110 | 1.6984 | 1.6826 | |
1.9664 | 1.9825 | 1.9652 | 1.9542 | 1.9375 | 1.9233 | 1.9053 | |
2.1859 | 2.1675 | 2.1558 | 2.1388 | 2.1223 | 2.1042 | ||
2.3741 | 2.3520 | 2.3437 | 2.3244 | 2.3089 | 2.2895 | ||
2.5270 | 2.5167 | 2.4970 | 2.4818 | 2.4627 | |||
2.6886 | 2.6807 | 2.6632 | 2.6467 | 2.6268 | |||
2.8359 | 2.8157 | 2.8016 | 2.7772 | ||||
2.9803 | 2.9638 | 2.9473 | 2.9238 | ||||
3.1035 | 3.0865 | 3.0624 | |||||
3.2351 | 3.2216 | 3.1945 | |||||
3.3497 | 3.3252 | ||||||
3.4718 | 3.4474 | ||||||
3.5652 | |||||||
3.6794 |
200 | 200 | 200 | 200 | 200 | 200 | 200 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5486 | 0.5318 | 0.5269 | 0.5207 | 0.5180 | 0.5145 | 0.5130 | |
0.4409 | 0.4557 | 0.4552 | 0.4586 | 0.4570 | 0.4570 | 0.4549 | |
0.8964 | 0.9173 | 0.9095 | 0.9112 | 0.9071 | 0.9070 | 0.9020 | |
1.2875 | 1.3083 | 1.3054 | 1.3049 | 1.2942 | 1.2925 | 1.2848 | |
1.6139 | 1.6366 | 1.6257 | 1.6265 | 1.6155 | 1.6105 | 1.6011 | |
1.8911 | 1.9201 | 1.9076 | 1.9136 | 1.9015 | 1.8981 | 1.8865 | |
2.1345 | 2.1715 | 2.1616 | 2.1657 | 2.1553 | 2.1488 | 2.1392 | |
2.3988 | 2.3878 | 2.3947 | 2.3837 | 2.3791 | 2.3673 | ||
2.6087 | 2.5963 | 2.6087 | 2.5930 | 2.5904 | 2.5795 | ||
2.7919 | 2.8033 | 2.7903 | 2.7884 | 2.7725 | |||
2.9716 | 2.9882 | 2.9766 | 2.9725 | 2.9578 | |||
3.1640 | 3.1468 | 3.1487 | 3.1324 | ||||
3.3281 | 3.3123 | 3.3123 | 3.2966 | ||||
3.4691 | 3.4709 | 3.4557 | |||||
3.6205 | 3.6227 | 3.6075 | |||||
3.7701 | 3.7512 | ||||||
3.9081 | 3.8927 | ||||||
4.0295 | |||||||
4.1614 |
300 | 300 | 300 | 300 | 300 | 300 | 300 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5490 | 0.5318 | 0.5266 | 0.5205 | 0.5180 | 0.5143 | 0.5129 | |
0.4534 | 0.4678 | 0.4666 | 0.4706 | 0.4696 | 0.4713 | 0.4696 | |
0.9374 | 0.9482 | 0.9445 | 0.9519 | 0.9486 | 0.9526 | 0.9496 | |
1.3456 | 1.3714 | 1.3611 | 1.3605 | 1.3508 | 1.3553 | 1.3496 | |
1.6910 | 1.7255 | 1.7138 | 1.7148 | 1.7049 | 1.7073 | 1.7001 | |
1.9880 | 2.0307 | 2.0157 | 2.0219 | 2.0084 | 2.0103 | 2.0058 | |
2.2470 | 2.2975 | 2.2822 | 2.2905 | 2.2776 | 2.2836 | 2.2728 | |
2.5398 | 2.5260 | 2.5360 | 2.5230 | 2.5290 | 2.5178 | ||
2.7641 | 2.7492 | 2.7598 | 2.7476 | 2.7539 | 2.7459 | ||
2.9559 | 2.9689 | 2.9586 | 2.9645 | 2.9548 | |||
3.1476 | 3.1676 | 3.1502 | 3.1638 | 3.1522 | |||
3.3505 | 3.3367 | 3.3494 | 3.3405 | ||||
3.5272 | 3.5136 | 3.5292 | 3.5158 | ||||
3.6829 | 3.6986 | 3.6892 | |||||
3.8398 | 3.8602 | 3.8491 | |||||
4.0177 | 4.0039 | ||||||
4.1662 | 4.1569 | ||||||
4.2997 | |||||||
4.4396 |
370 | 370 | 370 | 370 | 370 | 370 | 370 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5489 | 0.5316 | 0.5267 | 0.5208 | 0.5178 | 0.5144 | 0.5130 | |
0.4822 | 0.5002 | 0.4747 | 0.4773 | 0.4764 | 0.4785 | 0.4744 | |
0.9830 | 1.0085 | 0.9522 | 0.9539 | 0.9505 | 0.9545 | 0.9527 | |
1.4209 | 1.4516 | 1.3758 | 1.3710 | 1.3600 | 1.3616 | 1.3602 | |
1.7870 | 1.8226 | 1.7224 | 1.7251 | 1.7161 | 1.7210 | 1.7170 | |
2.1002 | 2.1490 | 2.0337 | 2.0357 | 2.0246 | 2.0302 | 2.0252 | |
2.3789 | 2.4335 | 2.3032 | 2.3066 | 2.2951 | 2.3016 | 2.2936 | |
2.6923 | 2.5494 | 2.5554 | 2.5423 | 2.5512 | 2.5462 | ||
2.9309 | 2.7765 | 2.7809 | 2.7690 | 2.7788 | 2.7749 | ||
2.9873 | 2.9948 | 2.9819 | 2.9936 | 2.9876 | |||
3.1838 | 3.1934 | 3.1799 | 3.1936 | 3.1881 | |||
3.3806 | 3.3685 | 3.3837 | 3.3782 | ||||
3.5565 | 3.5472 | 3.5600 | 3.5547 | ||||
3.7132 | 3.7353 | 3.7295 | |||||
3.8774 | 3.8995 | 3.8938 | |||||
4.0602 | 4.0521 | ||||||
4.2078 | 4.1982 | ||||||
4.3509 | |||||||
4.4898 |
400 | 400 | 400 | 400 | 400 | 400 | 400 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5486 | 0.5316 | 0.5267 | 0.5206 | 0.5180 | 0.5142 | 0.5129 | |
0.4935 | 0.5121 | 0.4817 | 0.4826 | 0.4785 | 0.4809 | 0.4794 | |
1.0109 | 1.0346 | 0.9718 | 0.9706 | 0.9607 | 0.9647 | 0.9619 | |
1.4632 | 1.4870 | 1.3973 | 1.3911 | 1.3722 | 1.3779 | 1.3723 | |
1.8388 | 1.8735 | 1.7536 | 1.7491 | 1.7297 | 1.7348 | 1.7287 | |
2.1609 | 2.2105 | 2.0728 | 2.0667 | 2.0436 | 2.0475 | 2.0394 | |
2.4470 | 2.5056 | 2.3480 | 2.3429 | 2.3170 | 2.3226 | 2.3141 | |
2.7729 | 2.6001 | 2.5929 | 2.5664 | 2.5744 | 2.5657 | ||
3.0145 | 2.8307 | 2.8272 | 2.7992 | 2.8061 | 2.7965 | ||
3.0449 | 3.0424 | 3.0084 | 3.0214 | 3.0094 | |||
3.2445 | 3.2425 | 3.2114 | 3.2261 | 3.2125 | |||
3.4314 | 3.3986 | 3.4169 | 3.4044 | ||||
3.6114 | 3.5831 | 3.6012 | 3.5891 | ||||
3.7542 | 3.7728 | 3.7589 | |||||
3.9180 | 3.9374 | 3.9242 | |||||
4.0984 | 4.0863 | ||||||
4.2527 | 4.2403 | ||||||
4.3870 | |||||||
4.5315 |
500 | 500 | 500 | 500 | 500 | 500 | 500 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5485 | 0.5314 | 0.5265 | 0.5208 | 0.5182 | 0.5142 | 0.5131 | |
0.5208 | 0.5440 | 0.5122 | 0.5059 | 0.4900 | 0.4909 | 0.4846 | |
1.0788 | 1.1047 | 1.0372 | 1.0241 | 0.9914 | 0.9933 | 0.9804 | |
1.5573 | 1.5953 | 1.4967 | 1.4720 | 1.4230 | 1.4251 | 1.4062 | |
1.9657 | 2.0181 | 1.8898 | 1.8564 | 1.7922 | 1.7919 | 1.7678 | |
2.3154 | 2.3803 | 2.2343 | 2.1982 | 2.1222 | 2.1218 | 2.0951 | |
2.6225 | 2.7034 | 2.5356 | 2.4945 | 2.4104 | 2.4119 | 2.3772 | |
2.9926 | 2.8089 | 2.7667 | 2.6718 | 2.6698 | 2.6350 | ||
3.2611 | 3.0592 | 3.0123 | 2.9119 | 2.9129 | 2.8743 | ||
3.2905 | 3.2440 | 3.1390 | 3.1376 | 3.1000 | |||
3.5081 | 3.4586 | 3.3519 | 3.3483 | 3.3045 | |||
3.6607 | 3.5419 | 3.5462 | 3.5036 | ||||
3.8516 | 3.7302 | 3.7389 | 3.6848 | ||||
3.9112 | 3.9227 | 3.8699 | |||||
4.0841 | 4.0928 | 4.0387 | |||||
4.2556 | 4.2060 | ||||||
4.4243 | 4.3662 | ||||||
4.5165 | |||||||
4.6654 |
600 | 600 | 600 | 600 | 600 | 600 | 600 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5486 | 0.5320 | 0.5268 | 0.5205 | 0.5181 | 0.5142 | 0.5132 | |
0.5482 | 0.5680 | 0.5379 | 0.5378 | 0.5144 | 0.5150 | 0.4981 | |
1.1295 | 1.1626 | 1.0953 | 1.0912 | 1.0411 | 1.0418 | 1.0075 | |
1.6341 | 1.6843 | 1.5875 | 1.5711 | 1.4977 | 1.4969 | 1.4460 | |
2.0644 | 2.1303 | 2.0031 | 1.9906 | 1.8963 | 1.8942 | 1.8294 | |
2.4385 | 2.5102 | 2.3621 | 2.3501 | 2.2444 | 2.2388 | 2.1667 | |
2.7623 | 2.8489 | 2.6882 | 2.6741 | 2.5530 | 2.5514 | 2.4636 | |
3.1570 | 2.9789 | 2.9677 | 2.8323 | 2.8265 | 2.7325 | ||
3.4369 | 3.2453 | 3.2338 | 3.0898 | 3.0840 | 2.9809 | ||
3.4938 | 3.4808 | 3.3257 | 3.3265 | 3.2161 | |||
3.7211 | 3.7120 | 3.5470 | 3.5446 | 3.4283 | |||
3.9335 | 3.7595 | 3.7615 | 3.6345 | ||||
4.1429 | 3.9641 | 3.9624 | 3.8278 | ||||
4.1480 | 4.1512 | 4.0177 | |||||
4.3312 | 4.3343 | 4.1938 | |||||
4.5149 | 4.3631 | ||||||
4.6828 | 4.5280 | ||||||
4.6861 | |||||||
4.8414 |
700 | 700 | 700 | 700 | 700 | 700 | 700 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5485 | 0.5318 | 0.5269 | 0.5208 | 0.5185 | 0.5146 | 0.5129 | |
0.5729 | 0.5981 | 0.5655 | 0.5654 | 0.5399 | 0.5408 | 0.5253 | |
1.1772 | 1.2201 | 1.1495 | 1.1459 | 1.0942 | 1.0956 | 1.0609 | |
1.7063 | 1.7644 | 1.6641 | 1.6521 | 1.5692 | 1.5707 | 1.5224 | |
2.1557 | 2.2336 | 2.1034 | 2.0897 | 1.9904 | 1.9890 | 1.9284 | |
2.5459 | 2.6385 | 2.4855 | 2.4705 | 2.3538 | 2.3501 | 2.2775 | |
2.8870 | 2.9966 | 2.8256 | 2.8117 | 2.6776 | 2.6768 | 2.5981 | |
3.3249 | 3.1313 | 3.1167 | 2.9722 | 2.9705 | 2.8835 | ||
3.6161 | 3.4151 | 3.4005 | 3.2427 | 3.2436 | 3.1445 | ||
3.6734 | 3.6624 | 3.4919 | 3.4926 | 3.3935 | |||
3.9141 | 3.9049 | 3.7263 | 3.7323 | 3.6206 | |||
4.1381 | 3.9465 | 3.9507 | 3.8318 | ||||
4.3565 | 4.1583 | 4.1654 | 4.0471 | ||||
4.3633 | 4.3671 | 4.2385 | |||||
4.5495 | 4.5640 | 4.4302 | |||||
4.7462 | 4.6082 | ||||||
4.9226 | 4.7855 | ||||||
4.9518 | |||||||
5.1086 |
800 | 800 | 800 | 800 | 800 | 800 | 800 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5487 | 0.5316 | 0.5269 | 0.5206 | 0.5181 | 0.5144 | 0.5134 | |
0.5900 | 0.6251 | 0.5884 | 0.5935 | 0.5703 | 0.5702 | 0.5459 | |
1.2147 | 1.2666 | 1.1924 | 1.1997 | 1.1524 | 1.1523 | 1.1024 | |
1.7631 | 1.8362 | 1.7274 | 1.7317 | 1.6618 | 1.6580 | 1.5869 | |
2.2318 | 2.3283 | 2.1874 | 2.1918 | 2.1045 | 2.0984 | 2.0080 | |
2.6372 | 2.7569 | 2.5894 | 2.5957 | 2.4860 | 2.4804 | 2.3743 | |
2.9928 | 3.1350 | 2.9422 | 2.9539 | 2.8334 | 2.8273 | 2.7040 | |
3.4745 | 3.2697 | 3.2773 | 3.1406 | 3.1403 | 3.0023 | ||
3.7846 | 3.5575 | 3.5746 | 3.4295 | 3.4231 | 3.2787 | ||
3.8327 | 3.8521 | 3.6982 | 3.6907 | 3.5340 | |||
4.0909 | 4.1112 | 3.9451 | 3.9418 | 3.7774 | |||
4.3554 | 4.1809 | 4.1773 | 3.9958 | ||||
4.5831 | 4.4028 | 4.4008 | 4.2094 | ||||
4.6160 | 4.6165 | 4.4223 | |||||
4.8154 | 4.8222 | 4.6167 | |||||
5.0125 | 4.8086 | ||||||
5.2057 | 4.9797 | ||||||
5.1611 | |||||||
5.3314 |
900 | 900 | 900 | 900 | 900 | 900 | 900 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5491 | 0.5318 | 0.5267 | 0.5204 | 0.5181 | 0.5148 | 0.5131 | |
0.6016 | 0.6433 | 0.6097 | 0.6173 | 0.5909 | 0.5892 | 0.5726 | |
1.2415 | 1.3066 | 1.2347 | 1.2472 | 1.1946 | 1.1904 | 1.1571 | |
1.8051 | 1.8973 | 1.7927 | 1.8051 | 1.7256 | 1.7173 | 1.6715 | |
2.2886 | 2.4069 | 2.2714 | 2.2870 | 2.1874 | 2.1766 | 2.1170 | |
2.7038 | 2.8523 | 2.6929 | 2.7108 | 2.5906 | 2.5776 | 2.5060 | |
3.0740 | 3.2459 | 3.0660 | 3.0866 | 2.9494 | 2.9373 | 2.8504 | |
3.5973 | 3.3982 | 3.4314 | 3.2756 | 3.2617 | 3.1673 | ||
3.9184 | 3.7021 | 3.7371 | 3.5752 | 3.5587 | 3.4569 | ||
3.9949 | 4.0333 | 3.8552 | 3.8354 | 3.7322 | |||
4.2626 | 4.3011 | 4.1154 | 4.0952 | 3.9796 | |||
4.5601 | 4.3648 | 4.3439 | 4.2216 | ||||
4.7992 | 4.5941 | 4.5710 | 4.4496 | ||||
4.8169 | 4.7958 | 4.6658 | |||||
5.0292 | 5.0140 | 4.8737 | |||||
5.2138 | 5.0763 | ||||||
5.4120 | 5.2603 | ||||||
5.4477 | |||||||
5.6263 |
1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | |
---|---|---|---|---|---|---|---|
6 | 8 | 10 | 12 | 14 | 16 | 18 | |
0.5488 | 0.5315 | 0.5267 | 0.5204 | 0.5180 | 0.5145 | 0.5131 | |
0.6210 | 0.6627 | 0.6288 | 0.6365 | 0.6123 | 0.6133 | 0.5929 | |
1.2830 | 1.3521 | 1.2773 | 1.2897 | 1.2401 | 1.2415 | 1.2002 | |
1.8660 | 1.9650 | 1.8565 | 1.8668 | 1.7946 | 1.7942 | 1.7343 | |
2.3673 | 2.4839 | 2.3538 | 2.3700 | 2.2772 | 2.2754 | 2.2008 | |
2.7970 | 2.9581 | 2.7936 | 2.8136 | 2.7032 | 2.6998 | 2.6098 | |
3.1839 | 3.3639 | 3.1819 | 3.2012 | 3.0771 | 3.0784 | 2.9711 | |
3.7312 | 3.5328 | 3.5546 | 3.4182 | 3.4149 | 3.3030 | ||
4.0695 | 3.8544 | 3.8793 | 3.7291 | 3.7279 | 3.6101 | ||
4.1454 | 4.1850 | 4.0249 | 4.0271 | 3.8824 | |||
4.4204 | 4.4694 | 4.2975 | 4.2991 | 4.1504 | |||
4.7359 | 4.5586 | 4.5521 | 4.4033 | ||||
4.9908 | 4.7935 | 4.8017 | 4.6367 | ||||
5.0253 | 5.0380 | 4.8617 | |||||
5.2440 | 5.2558 | 5.0762 | |||||
5.4759 | 5.2897 | ||||||
5.6774 | 5.4903 | ||||||
5.6778 | |||||||
5.8687 |
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C | SRC | C | SRC | C | SRC | |
---|---|---|---|---|---|---|
0.5 | 0.6428 | 0.5 | 0.6425 | 0.5 | 0.6428 | |
2.8497 | 0.798 | 4.3891 | 1.2031 | 5.0708 | 1.382 |
C | SRC | 6 | 8 | 10 | 12 | 14 | 16 | 18 | ||
---|---|---|---|---|---|---|---|---|---|---|
10 | DD | 5.5798 | 33.1970 | 28.9016 | 29.5565 | 34.0757 | 36.7935 | 40.1121 | 42.0636 | 45.7209 |
ARL | 100.1217 | 118.7456 | 99.9766 | 99.9764 | 99.6454 | 99.8677 | 99.9687 | 99.2931 | 99.6043 | |
FAR | 0.0655 | 0.0056 | 0 | 0 | 0 | 0.0001 | 0.0001 | 0.0002 | 0.0003 | |
20 | DD | 4.5829 | 12.4038 | 12.3431 | 12.7814 | 13.4354 | 14.7583 | 16.1362 | 16.9636 | 18.2698 |
ARL | 100.1221 | 118.7666 | 99.8431 | 100.1864 | 99.5062 | 100.0187 | 100.0108 | 99.4962 | 99.5444 | |
FAR | 0.1566 | 0.0607 | 0.0148 | 0.0084 | 0.0064 | 0.0280 | 0.0518 | 0.0642 | 0.0779 | |
30 | DD | 4.5789 | 8.2788 | 8.7000 | 9.3913 | 10.4685 | 11.2349 | 12.0707 | 12.4557 | 13.2802 |
ARL | 100.0206 | 118.8567 | 99.7885 | 100.2029 | 99.4369 | 100.1514 | 99.7156 | 99.4519 | 99.5012 | |
FAR | 0.2390 | 0.1306 | 0.0702 | 0.0875 | 0.1068 | 0.1164 | 0.1314 | 0.1403 | 0.1587 | |
40 | DD | 4.5749 | 6.9814 | 7.9473 | 8.5354 | 9.3278 | 9.8141 | 10.4779 | 10.9331 | 11.5477 |
ARL | 100.0611 | 118.8167 | 99.9674 | 100.3011 | 89.7641 | 100.0303 | 89.8406 | 99.3637 | 99.3303 | |
FAR | 0.3141 | 0.2013 | 0.1799 | 0.1818 | 0.1878 | 0.2029 | 0.2186 | 0.2268 | 0.2405 | |
50 | DD | 4.5802 | 6.3752 | 7.6181 | 7.9070 | 8.6649 | 9.2004 | 9.6475 | 10.0578 | 10.5800 |
ARL | 100.1531 | 118.8285 | 99.8177 | 100.2513 | 99.3827 | 99.9977 | 99.7287 | 99.4817 | 99.6277 | |
FAR | 0.3810 | 0.2695 | 0.2668 | 0.2620 | 0.2823 | 0.2889 | 0.2951 | 0.3072 | 0.3224 |
C | SRC | 6 | 8 | 10 | 12 | 14 | 16 | 18 | ||
---|---|---|---|---|---|---|---|---|---|---|
10 | DD | 7.4845 | 268.0231 | 127.1618 | 115.8420 | 114.1003 | 113.0386 | 118.7457 | 122.5208 | 134.0196 |
ARL | 500.3259 | 532.0541 | 489.4462 | 487.0666 | 483.9462 | 486.4092 | 500.5751 | 504.5539 | 527.5008 | |
FAR | 0.0088 | 0.0001 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
20 | DD | 7.4774 | 89.3780 | 26.3227 | 36.4466 | 27.1662 | 28.6116 | 30.5613 | 32.5018 | 35.2079 |
ARL | 499.8330 | 531.6094 | 484.7291 | 484.6379 | 484.0727 | 486.1253 | 500.1508 | 504.9982 | 526.9485 | |
FAR | 0.0280 | 0.0067 | 0.0012 | 0.0003 | 0.0001 | 0 | 0 | 0 | 0 | |
30 | DD | 7.4762 | 36.6425 | 15.4354 | 16.9657 | 17.7711 | 19.1151 | 20.2779 | 21.8232 | 23.2656 |
ARL | 500.0630 | 531.3130 | 489.2917 | 484.4814 | 484.2848 | 486.0503 | 499.7263 | 505.1260 | 526.9345 | |
FAR | 0.0473 | 0.0190 | 0.0094 | 0.0044 | 0.0030 | 0.0016 | 0.0010 | 0.0006 | 0.0003 | |
40 | DD | 7.4691 | 20.2862 | 13.0108 | 14.5117 | 15.2444 | 16.3110 | 17.1893 | 18.4266 | 19.5790 |
ARL | 500.2418 | 531.3190 | 487.2419 | 486.2570 | 484.3922 | 486.3174 | 500.3466 | 505.4857 | 527.5350 | |
FAR | 0.0661 | 0.0336 | 0.0233 | 0.0144 | 0.0117 | 0.0078 | 0.0056 | 0.0038 | 0.0025 | |
50 | DD | 7.4804 | 14.5257 | 12.0488 | 13.4507 | 14.0356 | 14.9430 | 15.6976 | 16.7346 | 17.6902 |
ARL | 499.8426 | 531.3030 | 487.4462 | 484.9996 | 483.3978 | 486.6009 | 500.5158 | 504.9296 | 527.0213 | |
FAR | 0.0849 | 0.0493 | 0.0410 | 0.0285 | 0.0252 | 0.0187 | 0.0147 | 0.0109 | 0.0077 |
C | SRC | 6 | 8 | 10 | 12 | 14 | 16 | 18 | ||
---|---|---|---|---|---|---|---|---|---|---|
10 | DD | 8.8165 | 648.1897 | 362.1763 | 324.2384 | 318.3756 | 317.0301 | 318.4782 | 316.5780 | 316.4286 |
ARL | 998.8042 | 1045.7143 | 1003.2 | 996 | 993.8 | 997 | 991.7 | 992.7 | 993.7 | |
FAR | 0.0030 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
20 | DD | 8.7880 | 254.2445 | 60.9372 | 54.4992 | 54.1869 | 57.0418 | 59.4549 | 62.6949 | 64.8553 |
ARL | 1003.3 | 1045.5714 | 1003.7 | 1001.1 | 993.1 | 996.5 | 992 | 992.8 | 993.8 | |
FAR | 0.0130 | 0.0023 | 0.0001 | 0 | 0 | 0 | 0 | 0 | 0 | |
30 | DD | 8.7839 | 102.8965 | 24.4908 | 25.4678 | 27.1288 | 29.9713 | 31.4247 | 34.1553 | 35.4113 |
ARL | 999.4 | 1044.6429 | 104.8 | 998.3 | 993.4 | 993.7 | 997.4 | 994.1 | 992.9 | |
FAR | 0.0229 | 0.0080 | 0.0021 | 0.0003 | 0.0002 | 0 | 0 | 0 | 0 | |
40 | DD | 8.7871 | 47.5679 | 17.5591 | 20.3572 | 21.7860 | 24.2974 | 25.4796 | 27.6644 | 28.7028 |
ARL | 1000.4571 | 1045.3143 | 1002.9 | 998.6 | 993.2 | 996.4 | 993.9 | 991.9 | 993.4 | |
FAR | 0.0328 | 0.0148 | 0.0061 | 0.0022 | 0.0014 | 0.0006 | 0.0003 | 0.0001 | 0.0001 | |
50 | DD | 8.7878 | 26.7629 | 15.8388 | 18.3891 | 19.6391 | 21.7982 | 22.8171 | 24.7035 | 25.5985 |
ARL | 1000.5 | 1045.7429 | 1007.6 | 997.3 | 994.2 | 996.9 | 995.7 | 993 | 993.3 | |
FAR | 0.0426 | 0.0222 | 0.0130 | 0.0065 | 0.0043 | 0.0023 | 0.0016 | 0.0009 | 0.0006 |
C | SRC | 6 | 8 | 10 | 12 | 14 | 16 | 18 | ||
---|---|---|---|---|---|---|---|---|---|---|
10 | DD | 3.8910 | 52.2870 | 46.3682 | 47.2927 | 52.3379 | 55.5305 | 59.8147 | 62.2322 | 66.5670 |
ARL | 24.7372 | 118.8468 | 99.6766 | 100.5088 | 99.6951 | 100.2877 | 99.7397 | 99.4211 | 99.5998 | |
FAR | 0.2834 | 0.0057 | 0 | 0 | 0 | 0 | 0.0001 | 0.0002 | 0.0004 | |
20 | DD | 3.8894 | 26.8525 | 23.5949 | 23.2869 | 24.8711 | 26.7932 | 29.3518 | 30.8005 | 33.2495 |
ARL | 24.7353 | 118.8606 | 100.0215 | 100.4114 | 99.8287 | 100.1349 | 99.7336 | 99.2912 | 99.6710 | |
FAR | 0.5369 | 0.0610 | 0.0147 | 0.0089 | 0.0066 | 0.0277 | 0.0529 | 0.0640 | 0.0778 | |
30 | DD | 3.8831 | 18.4651 | 15.8380 | 16.2014 | 18.1705 | 19.2565 | 20.6417 | 21.3250 | 22.9193 |
ARL | 24.7227 | 118.9065 | 99.9930 | 100.2403 | 99.7697 | 99.7791 | 99.8651 | 99.4605 | 99.5673 | |
FAR | 0.7007 | 0.1306 | 0.0696 | 0.0882 | 0.1088 | 0.1161 | 0.1334 | 0.1377 | 0.1585 | |
40 | DD | 3.9057 | 14.8526 | 13.8203 | 14.0883 | 15.1973 | 15.9524 | 17.1175 | 17.9106 | 18.9459 |
ARL | 24.7271 | 118.8372 | 99.9986 | 100.3971 | 99.7328 | 100.1732 | 99.6427 | 99.3361 | 99.6082 | |
FAR | 0.8071 | 0.2014 | 0.1821 | 0.1811 | 0.1890 | 0.2012 | 0.2189 | 0.2286 | 0.2415 | |
50 | DD | 3.8800 | 13.0198 | 12.6930 | 12.7257 | 13.9105 | 14.5288 | 15.3389 | 15.9883 | 16.9118 |
ARL | 24.7468 | 118.8856 | 99.9125 | 100.5026 | 99.7612 | 100.0074 | 99.6560 | 99.5814 | 99.4769 | |
FAR | 0.8756 | 0.2697 | 0.2665 | 0.2649 | 0.2843 | 0.2876 | 0.2937 | 0.3087 | 0.3216 |
C | SRC | 6 | 8 | 10 | 12 | 14 | 16 | 18 | ||
---|---|---|---|---|---|---|---|---|---|---|
10 | DD | 6.3947 | 353.9665 | 225.9275 | 209.4729 | 205.9059 | 204.0480 | 213.5904 | 214.6237 | 233.6764 |
ARL | 65.5479 | 531.4316 | 488.0331 | 484.4087 | 488.4807 | 488.2045 | 499.9793 | 506.4905 | 526.8804 | |
FAR | 0.0931 | 0.0001 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
20 | DD | 6.3823 | 201.6783 | 81.3509 | 71.7558 | 70.5015 | 70.3994 | 72.9243 | 75.0408 | 82.1264 |
ARL | 65.5938 | 531.5086 | 488.6755 | 483.7614 | 485.5939 | 487.7366 | 501.0609 | 506.4544 | 527.0465 | |
FAR | 0.2273 | 0.0068 | 0.0015 | 0.0003 | 0.0001 | 0 | 0 | 0 | 0 | |
30 | DD | 6.3845 | 123.0448 | 40.4024 | 37.0531 | 36.8415 | 37.7073 | 39.4376 | 41.2816 | 44.6164 |
ARL | 65.6212 | 531.4936 | 488.4589 | 482.7787 | 485.7063 | 487.1630 | 488.9567 | 507.7213 | 526.9891 | |
FAR | 0.3423 | 0.0192 | 0.0100 | 0.0043 | 0.0029 | 0.0017 | 0.0010 | 0.0005 | 0.0004 | |
40 | DD | 6.3847 | 81.4736 | 27.3471 | 26.5232 | 27.0461 | 28.0854 | 29.3648 | 30.9860 | 33.1040 |
ARL | 65.5715 | 531.2075 | 488.4647 | 485.5648 | 485.1457 | 487.8131 | 500.0233 | 506.9858 | 527.3921 | |
FAR | 0.4401 | 0.0338 | 0.0245 | 0.0152 | 0.0114 | 0.0080 | 0.0056 | 0.0036 | 0.0023 | |
50 | DD | 6.3891 | 57.7347 | 22.3869 | 22.5064 | 23.1674 | 24.1505 | 25.1836 | 26.6830 | 28.3272 |
ARL | 65.5934 | 531.5276 | 488.1271 | 484.9405 | 485.9766 | 487.5572 | 498.3469 | 506.6913 | 526.9864 | |
FAR | 0.5233 | 0.0493 | 0.0417 | 0.0297 | 0.0250 | 0.0191 | 0.0149 | 0.0108 | 0.0076 |
C | SRC | 6 | 8 | 10 | 12 | 14 | 16 | 18 | ||
---|---|---|---|---|---|---|---|---|---|---|
10 | DD | 7.5806 | 794.075 | 571.8691 | 533.4575 | 521.0822 | 520.3675 | 519.0203 | 508.9829 | 512.4341 |
ARL | 97.3143 | 1044.2857 | 1007.3 | 996.5 | 995.5 | 998.7 | 997.2 | 991.8 | 992.1 | |
FAR | 0.0554 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
20 | DD | 7.5627 | 509.1507 | 223.2311 | 187.2751 | 180.3638 | 179.9559 | 175.3016 | 177.2089 | 178.9137 |
ARL | 97.3286 | 1044.3714 | 1008.3 | 997.5 | 994.8 | 999.6 | 997.5 | 995.3 | 997.6 | |
FAR | 0.1511 | 0.0023 | 0.0001 | 0 | 0 | 0 | 0 | 0 | 0 | |
30 | DD | 7.5561 | 331.2999 | 93.7481 | 76.7901 | 75.5415 | 75.9972 | 76.4086 | 78.6915 | 80.2621 |
ARL | 97.3571 | 1044.9714 | 1006.5 | 996.5 | 998.6 | 1002.3 | 997.9 | 990.1 | 997.2 | |
FAR | 0.2375 | 0.0079 | 0.0018 | 0.0004 | 0.0002 | 0 | 0 | 0 | 0 | |
40 | DD | 7.5618 | 221.1639 | 50.4663 | 44.4734 | 44.9347 | 47.2327 | 48.3099 | 51.1874 | 52.2784 |
ARL | 97.3 | 1044.4857 | 1006.6 | 996.6 | 994.7 | 998.5 | 997.2 | 994.3 | 994.9 | |
FAR | 0.3152 | 0.0149 | 0.0063 | 0.0025 | 0.0015 | 0.0006 | 0.0003 | 0.0002 | 0.0001 | |
50 | DD | 7.5575 | 152.8864 | 35.1211 | 33.7103 | 34.8390 | 37.2713 | 38.4121 | 41.0106 | 42.1451 |
ARL | 97.3429 | 1044.6571 | 1006.7 | 996.2 | 994.1 | 996.6 | 996.8 | 992.8 | 995.1 | |
FAR | 0.3848 | 0.0224 | 0.0131 | 0.0064 | 0.0043 | 0.0023 | 0.0016 | 0.0009 | 0.0007 |
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Lang, M. Control Limits for an Adaptive Self-Starting Distribution-Free CUSUM Based on Sequential Ranks. Technologies 2019, 7, 71. https://doi.org/10.3390/technologies7040071
Lang M. Control Limits for an Adaptive Self-Starting Distribution-Free CUSUM Based on Sequential Ranks. Technologies. 2019; 7(4):71. https://doi.org/10.3390/technologies7040071
Chicago/Turabian StyleLang, Michael. 2019. "Control Limits for an Adaptive Self-Starting Distribution-Free CUSUM Based on Sequential Ranks" Technologies 7, no. 4: 71. https://doi.org/10.3390/technologies7040071